Massive and massless plasmons in germanene nanosheets

Atomically thin crystals may exhibit peculiar dispersive electronic states equivalent to free charged particles of ultralight to ultraheavy masses. A rare coexistence of linear and parabolic dispersions yields correlated charge density modes exploitable for nanometric light confinement. Here, we use a time-dependent density-functional approach, under several levels of increasing accuracy, from the random-phase approximation to the Bethe-Salpeter equation formalism, to assess the role of different synthesized germanene samples as platforms for these plasmon excitations. In particular, we establish that both freestanding and some supported germenene monolayers can sustain infrared massless modes, resolved into an out-of-phase (optical) and an in-phase (acoustic) component. We further indicate precise experimental geometries that naturally host infrared massive modes, involving two different families of parabolic charge carriers. We thus show that the interplay of the massless and massive plasmons can be finetuned by applied extrinsic conditions or geometry deformations, which constitutes the core mechanism of germanene-based optoelectronic and plasmonic applications.

(a) LDA Band dispersions along the ΓKMΓ border of the irreducible 1 st BZ of FGe, decomposed according to their σ (s and in-plane p atomic orbitals), π (out-of-plane p atomic orbitals) and d characters. (b) DOS components of FGe, related to the projected bands in (a), and total DOS resulting from the full band dispersions of FGe, reported in Fig. 1(a) of the main text. The highest valence and lowest conduction bands have dominant π and π * characters, respectively, over a wide region around the Dirac cone (with vertex E C = E F at K). This region extends up to the flat parts of the two bands (at M ), where the π-DOS and π * -DOS display sharp peaks, associated to the VHSs. Within the same region, the second-highest valence and second-lowest conduction bands have dominant σ and σ * characters, respectively. Unlike graphene and like silicene, the in-plane mirror symmetry is broken by structural buckling, which allows for sp 2 (σ, σ * ) to sp 3 (σ-π, σ * -π * ) hybridization changes within the same band (gray boxes). As a result, the π-like band becomes σ-like around Γ, sharing the same character with the second-highest valence band. Additionally, the two highest valence bands tend to the degenerate value E Γ σ at Γ. Complementary, the π * and σ * characters of the lowest two conduction bands respectively switch to σ * and π * , around Γ, with an avoided crossing point along ΓM (cyan box). Finally, a d-like character is recorded in the unoccupied bands at energies larger than ∼3 eV above E F . Nonetheless, the d-DOS has a small non-negligible effect on the occupied bands, starting from ∼-2 eV below E F .   Figure S3 (a) LDA Band dispersions along ΓKMΓ and (b) related DOS of QFGe on MoS 2 , decomposed as in Figs. S1 and S2, with same labels and shadings. We notice an overall similarity of the bands and DOS components with FGe ( Fig. S1) and QFGe on AlN (Fig. S2). Nonetheless, the two highest valence bands of this QFGe structure are further shifted up in energy, in such a way that the σ-like bands cross the Fermi level close to the Γ point, offering a bunch of unoccupied states just above E F . As a result, this QFGe structure is a metal, with E C shifted below E F . In addition, the σ * -like band lies above the π * -like band.
of (a) FGe, QFGe on AlN, and (b) QFGe on MoS 2 , computed with the same input parameters as the absorption spectrum of Fig. 3 in the main text. In particular, the minimum probing momentum allowed by the 720×720×1 MP-grid was applied, as detailed in the section "Time-dependent density functional approach" of the main text. The jDOS provides complementary information on the leading interband SPEs that contribute to the absorption peaks at L-IR to M-UV wavelengths. As shown in Figs. S5-S7 below, these excitations involve the highest or second highest occupied σ-and π-like states and the lowest and second lowest unoccupied σ-, σ * -, and π * -like states. In particular, the L-IR to S-IR feature in QFGe on MoS 2 is due to SPEs between the two σ-like bands at the crossing point with the Fermi level of the metal, and corresponds the shoulder at 0.45 eV in the absorption lineshape of Fig. 3  . These processes, which connect initial and final energy levels with high DOS, include: σ-σ * SPEs around Γ, at 1.18 eV (NIR); π-π * SPEs around M, at 1.80 eV (VIS); σ-π * SPEs close to Γ, and, to a minor extent, σ-σ * SPEs around the mid points of the ΓK and MΓ segments, at 3.10 eV (VIS-NUV); σ-π * SPEs, around the mid points of the ΓK and MΓ segments, and σ-σ * SPEs around M, at 3.37 eV (NUV); π-σ * SPEs around the mid point of the KM segment, at 3.76 eV (NUV). σ-σ * SPEs along the ΓK and MΓ segments, around Γ at 4.18 eV (MUV). abs. peaks σ* σ* Figure S6 (a) Energy bands, along the high-symmetry ΓKMΓ contour of the IBZ, (b) density of states, and (a), (b) dominant vertical transitions contributing to the NIR-NUV peak structures in the absorption and jDOS spectra of QFGe on AlN, which include: σ-σ * SPEs around Γ, at 1.40 eV (NIR); π-π * SPEs around M, at 1.80 eV (VIS); σ-π * SPEs, and, to a minor extent, σ-σ * SPEs along the along the ΓK and MΓ lines, towards Γ at 3.10 eV (VIS-NUV). σ-π * SPEs, along the ΓK and MΓ segments towards Γ, and σ-σ * SPEs around M, at 3.42 eV (NUV); σ-σ * SPEs along the ΓK and MΓ segments, around Γ at 4.13 eV (MUV). (a) Energy bands, along the high-symmetry ΓKMΓ contour of the IBZ, (b) density of states, and (a), (b) dominant vertical transitions contributing to the NIR-NUV peak structures in the absorption and jDOS spectra of QFGe on MoS 2 , which include: σ-σ SPEs around Γ, at 0.45 eV (MIR-NIR); σ-σ * SPEs around Γ, at 1.54 eV (NIR); π-π * SPEs around M, at 1.80 eV (VIS); σ-π * SPEs, and, to a minor extent, σ-σ * SPEs along the ΓK and MΓ lines, around their mid points and Γ at 2.63 eV (VIS); σ-π * SPEs around the mid points of the ΓK and MΓ lines, at 3.21 eV (NUV-VIS); π-σ * SPEs around M, at 3.38 eV (NUV); σ-π * SPEs along the ΓK and MΓ segments, towards Γ, and σ-σ * SPEs along the MΓ segment, towards M, at 3.47 eV (NUV); π-σ * SPEs around the mid points of the ΓK and MΓ lines, and σ-σ * SPEs, along ΓK, closer to Γ, at 4.12 eV.    Fig. S9 shows an AP mode, which is absent in the q∥ΓM-case. The same feature has also been found in graphene and silicene 37,38 . All other settings are as in Fig. S9.

Figure S12
Loss function of QFGe on AlN for the extrinsic conditions ∆E F =±0.2 and momentum transfers (a), (b) q∥ΓM, (c), (d) q∥ΓK, which lead to a scenario typical of 2DDMs with group IV atomic elements, including FGe in the LDA geometry (see Figs. S9, S10 above). This oscillation of the massless charge carriers can be individually tuned by changing the Fermi level position in the range where the σ-like or σ * -like states are left inert, namely −0.31≤∆E F ≤1.00 eV, as also suggested by the associated band dispersions (see Fig. 1b of the main text). The dominant 2DP propagates mostly undamped at MIR to NIR energies, and a small AP is activated by an applied momentum q∥ΓK.  Fig. 7(b), with q∥ΓM in spite of q∥ΓK, as attested by the absence or presence of the AP. In both the q∥ΓK and q∥ΓM cases, the Fermi level is driven around the top of the σ states, which generates a V-shaped massive mode on the same energy range as the 2DP, but shifted in momentum space. The intensity color scale is the same as in Fig. 7

Figure S14
Loss function of QFGe on MoS 2 for the extrinsic condition ∆E F = −0.29 eV that drives the Fermi level at the Dirac cone vertex. A complementary view of Fig. 5(c) and Fig. 6a of the main text is offered, which provides a complete analysis of the intraband σP and σ ′ P for q∥ΓK and q∥ΓM. In the shifted loss spectra of (c) and (d), a tiny 2DP is barely visible at energies similar to the corresponding excitation in intrinsic FGe [ Fig. 5(a)] and QFGe on AlN [ Fig. 5(b)].

Figure S15
Loss function of QFGe on MoS 2 for (a) ∆E F = 0.075 eV, (b) ∆E F = 0.09 eV and q∥ΓK. As in Fig. 7(a), (b) of the main text, the Fermi level is driven close to the top of the σ-like band, which causes the appearance of a V-shaped massive plasmon disjoint from the 2DP and shifted in momentum space. All other settings are as in Figs. S10, S12, S13, S14(a),(b).